All atomic nuclei of elements with an odd atomic mass or an odd atomic number possess a nuclear magnetic moment. Nuclear magnetic resonance is a phenomenon exhibited by this select group of atomic nuclei (termed "NMR active" nuclei), and is based upon the interaction of the nucleus with an applied, external magnetic field. The magnetic properties of a nucleus are conveniently discussed in terms of two quantities: the gyromagnetic ratio (.gamma.); and the nuclear spin (I). When an NMR active nucleus is placed in a magnetic field, its nuclear magnetic energy levels are split in to (2I+1) non-degenerate energy levels, which are separated from each other by an energy difference that is directly proportional to the strength of the applied magnetic field. This splitting is called the "Zeeman" splitting and the energy difference is equal to .gamma.hH.sub.0 /2.pi., where h is Plank's constant and H.sub.0 is the strength of the applied magnetic field. The frequency corresponding to the energy of the Zeeman splitting (.omega..sub.0 =.gamma.H.sub.0) is called the "Larmor frequency" and is proportional to the field strength of the magnetic field. Typical NMR active nuclei include .sup.1 H (protons), .sup.13 C, .sup.19 F, and .sup.31 P. For these four nuclei I=1/2, and each nucleus has two nuclear magnetic energy levels.
When a bulk sample of material containing NMR active nuclei is placed within a magnetic field called the main static field, the nuclear spins distribute themselves amongst the nuclear magnetic energy levels in accordance with Boltzmann's statistics. This results in a population imbalance among the energy levels and a net nuclear magnetization. It is this net nuclear magnetization that is studied by NMR techniques.
At equilibrium, the net nuclear magnetization of the aforementioned bulk sample is aligned parallel to the external magnetic field and is static (by convention, the direction of the main static field is taken to be the z-axis). A second magnetic field perpendicular to the main static magnetic field and rotating at, or near, the Larmor frequency can be applied to induce a coherent motion of the net nuclear magnetization. Since, at conventional main static magnetic field strengths, the Larmor frequency is in the megahertz frequency range, this second magnetic field is called a "radio frequency" or RF field.
The effect of the RF field is to shift the nuclear magnetization direction so that it is no longer parallel to the main static field. This shift introduces a net coherent motion of the nuclear magnetization about the main static field direction called a "nutation". In order to conveniently deal with this nutation, a reference frame is used which rotates about the laboratory reference frame z-axis at the Larmor frequency and also has its z-axis parallel to the main static field direction. In this "rotating frame" the net nuclear magnetization, which is rotating in the stationary "laboratory" reference frame, is now static.
Consequently, the effect of the RF field is to rotate the now static nuclear magnetization direction at an angle with respect to the main static field direction (z-axis). The new magnetization direction can be broken into a component which is parallel to the main field direction (z-axis direction) and a component which lies in the plane transverse to the main magnetization (x,y plane). The RF field is typically applied in pulses of varying length and amplitude and, by convention, an RF pulse of sufficient amplitude and length to rotate the nuclear magnetization in the rotating frame through an angle of 90.degree., or .pi./2 radians, and entirely into the x,y plane is called a ".pi./2 pulse".
Because the net nuclear magnetization is rotating with respect to the laboratory frame, the component of the nuclear magnetization that is transverse to the main magnetic field or that lies in the x,y plane rotates about the external magnetic field at the Larmor frequency. This rotation can be detected with a receiver coil that is resonant at the Larmor frequency. The receiver coil is generally located so that it senses voltage changes along one axis (for example, the x-axis) where the rotating magnetization component appears as an oscillating voltage. Frequently, the "transmitter coil" employed for applying the RF field to the sample and the "receiver coil" employed for detecting the magnetization are one and the same coil.
Although the main static field is applied to the overall material sample, the nuclear magnetic moment in each nucleus within the sample actually experiences an external magnetic field that is changed from the main static field value due to a screening from the surrounding electron cloud. This screening results in a slight shift in the Larmor frequency for that nucleus (called the "chemical shift" since the size and symmetry of the shielding effect is dependent on the chemical composition of the sample).
In a typical NMR experiment, the sample is placed in the main static field and a .pi./2 pulse is applied to shift the net magnetization into the transverse plane (called transverse magnetization). After application of the pulse, the transverse magnetization, or "coherence", begins to precess about the x-axis, or evolve, due to the chemical shifts at a frequency which is proportional to the chemical shift field strength. In the rotating frame, the detector (which is stationary in the laboratory frame) appears to rotate at the Larmor frequency. Consequently, the detector senses an oscillation produced by an apparent magnetization rotation at a frequency which is proportional to the frequency difference between the Larmor frequency and the chemical shift frequency.
Thus, the detected signal oscillates at the frequency shift difference. In addition to precessing at the Larmor frequency, in the absence of the applied RF field energy, the nuclear magnetization also undergoes two spontaneous processes: (1) the precessions of various individual nuclear spins which generate the net nuclear magnetization become dephased with respect to each other so that the magnetization within the transverse plane loses phase coherence (so-called "spin-spin relaxation") with an associated relaxation time, T.sub.2, and (2) the individual nuclear spins return to their equilibrium population of the nuclear magnetic energy levels (so-called "spin-lattice relaxation") with an associated relaxation time, T.sub.1. The latter process causes the received signal to decay to zero. The decaying, oscillating signal is called a free induction decay (FID).
Although many NMR experiments are designed such that the spin dynamics are uniform through the sample, there are cases where it is advantageous to impose a spatial heterogeneity across the sample. Some examples include imaging experiments, diffusion experiments, coherence transfer experiments where the heterogeneity may be used as a means of allowing a variation in coherence pathways, and multiple-quantum filtering experiments.
All of these latter experiments can be performed with a spatially-varying magnetic field. Some common examples of such a spatially-varying field include B.sub.0 and B.sub.1 magnetic field gradients which are gradients along the direction of the main static field and in the plane transverse to the main static field direction, respectively. Another spatially-varying field is the field associated with a "radial pulse". A "radial" RF pulse is a pulse that has a uniform RF field strength throughout the sample and a phase (relative to the detection coil phase) with a spatial dependence such that all possible phase differences are equally represented throughout the sample. Such a pulse is described in more detail in a copending patent application entitled "Method for Improving Selectivity in NMR Experiments Involving Coherence Transformation", filed on Mar. 12, 1993 by David G. Cory, Frank H. Laukien and Werner E. Maas and assigned to the same assignee as the present invention, which application is hereby incorporated by reference.
The application of an RF radial field to the sample corresponds to inducing spin evolution about an axis in the transverse plane, and it is the phase angle of this axis of rotation that varies across the sample.
For certain experiments (such as multiple quantum filters) it is advantageous to apply a spatially-varying rotation about the z-axis. As used in the discussion below, the word "coherence" describes a transition between different energy levels. The transitions are characterized by a change .DELTA.m.sub.z in the quantum number, m.sub.z and the change .DELTA.m.sub.z is called the coherence number. In any given system, only certain changes or coherence numbers are possible in accordance with quantum theory. If an RF pulse sequence is applied to a spin system, many of the possible coherences occur under the influence of RF pulses, but only those in which the change .DELTA.m.sub.z is .+-.1 produce observable magnetization in accordance with the "selection rule" for NMR spectroscopy.
A multiple quantum filtering experiment is based on the simple idea that an .omega.I.sub.z interaction drives an n-quantum coherence at a rate of n.omega. about the z-axis and takes advantage of the NMR selection rule. For example, if a sample is placed in a multiple quantum state involving single, double and triple quantum coherences and then a spatially-varying z rotation is applied, the triple quantum term will evolve three times as far as the single quantum term and the double quantum will evolve twice as far as the single quantum term. However, all coherences will be dephased.
If the multiple-quantum coherences are then converted into single quantum coherences (that is, observable magnetization) they will still be dephased. The application of a gradient equal to -1 times the original gradient will refocus the coherences that evolved through a single quantum state. Similarly, the application of a gradient equal to -2 times the original gradient will refocus the coherences that evolved through a double quantum state, and, finally, the application of a gradient equal to -3 times the original gradient will refocus the coherences that evolved through a triple quantum state. Thus, a filter for a specific coherence transformation can be constructed.
RF pulse sequences that introduce a spatially-varying z rotation are known in relationship to B.sub.0 and B.sub.1 gradients, but no such pulse configuration is known for radial pulses. Accordingly, it is an object of the present invention to provide a pulse sequence that converts a radial RF pulse into a spatially varying z rotation.